LIE-ALGEBRAS AND POLYNOMIALS IN ONE VARIABLE

A classification of linear differential and difference equations in one variable having polynomial solutions (the generalized Bochner problem) is presented. The idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2(R) (for differential equations) or 912(R)q (for finite-difference equations) given in the finite-dimensional 'projectivized' representation. Connection with the recently discovered quasi-exactly-solvable Schrodinger equations is discussed.